Strata and stabilizers of trees Vincent Guirardel Joint work with - - PowerPoint PPT Presentation

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Strata and stabilizers of trees

Vincent Guirardel Joint work with G. Levitt

Institut de Math´ ematiques de Toulouse

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Goal of the talk

Outer space CV

N =

• minimal free actions of FN on simplicial

trees

• / ∼.

Compactification CV

N =

• minimal very small actions on

R-trees

• / ∼.

Main example: action with trivial arc stabilizers. Goal Given T ∈ CV

N, find some structure that more or less parallels the

strata of a relative train track map.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Goal of the talk

Outer space CV

N =

• minimal free actions of FN on simplicial

trees

• / ∼.

Compactification CV

N =

• minimal very small actions on

R-trees

• / ∼.

Main example: action with trivial arc stabilizers. Goal Given T ∈ CV

N, find some structure that more or less parallels the

strata of a relative train track map. Applications Give some kind of decomposition of any T ∈ CV

N into simple

building blocks. Understand the stabilizer of T in Out(FN).

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

An example

α automorphism of a, b, c, d: α :            a → ab b → bab

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

An example

α automorphism of a, b, c, d: α :            c → d d → cad a → ab b → bab

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

An example

α automorphism of a, b, c, d: α :            c → d d → cad a → ab b → bab #{c, d} ≈ (1.6)k ∧ #{a, b} ≈ (2.6)k successive images of d:

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

An example

α automorphism of a, b, c, d: α :            c → d d → cad a → ab b → bab

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

An example

α automorphism of a, b, c, d: α :            c → d d → cad a → ab b → bab successive images of the path d, rescaled by 2.6k

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Tree interpretation: axis of the element d on

1 2.6k T.αk.

At the limit: FN acts on some R-tree T∞.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Tree interpretation: axis of the element d on

1 2.6k T.αk.

At the limit: FN acts on some R-tree T∞. Facts T∞ is α-invariant: there exists an α-equivariant homothety Hα : T∞ → T∞ a, b preserves a subtree Y ⊂ T∞, Y is Hα-invariant. Y is closed and disjoint from its translates

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

One can collapse Y equivariantly and get a topological R-tree, with an action of FN: Y ֒ → T∞ ։ T/Y

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Other description of the collapsed tree: T/Y = T ∞. Collapse all red edges before taking limit:

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Theorem [G-Levitt] Any T ∈ CV

N can be obtained from simplicial trees and mixing

trees by iterating two constructions: extensions Y ֒ → T ։ T/Y graph of actions Mixing: minimality condition ⇒ every orbits meets every segment in a dense set. Graph of actions = Free amalgamated product of actions on R-trees, glued along points.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Theorem [G-Levitt] Any T ∈ CV

N can be obtained from simplicial trees and mixing

trees by iterating two constructions: extensions Y ֒ → T ։ T/Y graph of actions Remark: this obliges to consider topological R-trees, with (non-nesting) actions by homeomorphisms. If mixing, such topological actions have an invariant metric.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Admissible subtrees

To simplify, assume T has no simplicial arc (branch points are dense), arc stabilizers are trivial. Definition A subtree Y ⊂ T is admissible if Y is not a point and any two distinct translates of Y are disjoint.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Admissible subtrees

To simplify, assume T has no simplicial arc (branch points are dense), arc stabilizers are trivial. Definition A subtree Y ⊂ T is admissible if Y is not a point and any two distinct translates of Y are disjoint. Example 1. Y ⊂ T∞ above.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Admissible subtrees

To simplify, assume T has no simplicial arc (branch points are dense), arc stabilizers are trivial. Definition A subtree Y ⊂ T is admissible if Y is not a point and any two distinct translates of Y are disjoint. Example 1. Y ⊂ T∞ above. Example 2. If T is simplicial, Y admissible ⇔ Y subgraph of groups A0 ∗C1 A1 ∗C2 A2

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Admissible subtrees

To simplify, assume T has no simplicial arc (branch points are dense), arc stabilizers are trivial. Definition A subtree Y ⊂ T is admissible if Y is not a point and any two distinct translates of Y are disjoint. Example 1. Y ⊂ T∞ above. Example 2. If T is simplicial, Y admissible ⇔ Y subgraph of groups A0 ∗C1 A1 ∗C2 A2 Example 3. T is mixing if and only if it has no admissible subtree.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Main finiteness result

Main finiteness result [G-Levitt] There are only finitely many orbits of admissible subtrees Y ⊂ T. For each admissible Y , ∂Y consists of finitely many orbits.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Main finiteness result

Main finiteness result [G-Levitt] There are only finitely many orbits of admissible subtrees Y ⊂ T. For each admissible Y , ∂Y consists of finitely many orbits. Next goal Use this theorem to understand the Out(FN)-stabilizer of T. Projective stabilizer Aut([T]) = set of α ∈ Aut(FN) s.t. ∃ α-equivariant homothety Hα : T → T. Isometric stabilizer: Aut(T) = set of α ∈ Aut(FN) s.t. ∃ α-equivariant isometry Hα : T → T. Out([T]) and Out(T) = their images in Out(FN).

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Stabilizer of a simplicial tree

Γ a graph of groups, T = ˜ Γ Bass-Serre tree. General facts:

1 Out0(˜

Γ) ⊂ Out(˜ Γ) finite index subgroup acting trivially on Γ.

2 There is a map ρ : Out0(˜

Γ) →

v Out(Gv)

3 Dehn twists are in the kernel of ρ 4 Elements of Out(Gv) which act like a conjugation on each

edge group are in the image of ρ Def: McCool group Fix {E1, . . . En} some subgroups in free group Fk. The set of automorphisms α ∈ Out(Fk) acting like a conjugation on each Ei is a McCool group.

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Theorem (G-Levitt) Fix T ∈ CV

N.

Out(T) has a finite index subgroup Out0(T) s.t. 1 →

• free groups → Out0(T) →
• McCool gps → 1

The set of scaling factors of Out([T]) is a cyclic subgroup of R∗

+ [Lustig]

Remark: the McCool groups are McCool groups of point

• stabilizers. The free groups correspond to Dehn twists.

Proposition McCool groups virtually have a finite classifying space. Corollary So does the stabilizer of T in Out(FN).

Vincent Guirardel, Toulouse Strata and stabilizers of trees

Goal of the talk An example Admissible subtrees Stabilizers Stabilizers Proof

Proof

Idea: construct a simplicial tree ˜ Γ on which Out(T) acts.

1 All automorphisms α in some finite index subgroup of

Out0(T) ⊂ Out(T) are piecewise-FN.

2 Out0(T) is uniformly piecewise-FN: there exists a piecewise

decomposition of T that is compatible with every α ∈ Out0(T).

3 There is a simplicial tree ˜

Γ dual to this piecewise decomposition

4 Out0(T) occurs as an extension of McCool groups by Dehn

twists in Γ.

Vincent Guirardel, Toulouse Strata and stabilizers of trees